ITP001 Axioms: ITP050^7.ax
%------------------------------------------------------------------------------
% File : ITP050^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : quotient.ax [Gau19]
% : HL4050^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 137 ( 26 unt; 31 typ; 0 def)
% Number of atoms : 273 ( 57 equ; 3 cnn)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 2022 ( 3 ~; 3 |; 57 &;1746 @)
% ( 24 <=>; 189 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 13 avg;1746 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 638 ( 638 >; 0 *; 0 +; 0 <<)
% Number of symbols : 32 ( 30 usr; 1 con; 0-8 aty)
% Number of variables : 826 ( 16 ^ 768 !; 2 ?; 826 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Equotient_2E_2D_2D_3E,type,
c_2Equotient_2E_2D_2D_3E:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27c ) > ( A_27b > A_27d ) > ( A_27c > A_27b ) > A_27a > A_27d ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Equotient_2E_3D_3D_3D_3E,type,
c_2Equotient_2E_3D_3D_3D_3E:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27b > A_27b > $o ) > ( A_27a > A_27b ) > ( A_27a > A_27b ) > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Equotient_2E_3F_21_21,type,
c_2Equotient_2E_3F_21_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ecombin_2EC,type,
c_2Ecombin_2EC:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > A_27b > A_27a > A_27c ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Equotient_2EEQUIV,type,
c_2Equotient_2EEQUIV:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Ecombin_2EK,type,
c_2Ecombin_2EK:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > A_27a ) ).
thf(c_2Ebool_2ELET,type,
c_2Ebool_2ELET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Equotient_2EPARTIAL__EQUIV,type,
c_2Equotient_2EPARTIAL__EQUIV:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Equotient_2EQUOTIENT,type,
c_2Equotient_2EQUOTIENT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > A_27b ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Ebool_2ERES__ABSTRACT,type,
c_2Ebool_2ERES__ABSTRACT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ebool_2ERES__EXISTS,type,
c_2Ebool_2ERES__EXISTS:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Equotient_2ERES__EXISTS__EQUIV,type,
c_2Equotient_2ERES__EXISTS__EQUIV:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ERES__EXISTS__UNIQUE,type,
c_2Ebool_2ERES__EXISTS__UNIQUE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ERES__FORALL,type,
c_2Ebool_2ERES__FORALL:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ecombin_2EW,type,
c_2Ecombin_2EW:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ebool_2Eliteral__case,type,
c_2Ebool_2Eliteral__case:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).
thf(c_2Equotient_2Erespects,type,
c_2Equotient_2Erespects:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Equotient_2EEQUIV__def,axiom,
! [A_27a: $tType,V0E: A_27a > A_27a > $o] :
( ( c_2Equotient_2EEQUIV @ A_27a @ V0E )
<=> ! [V1x: A_27a,V2y: A_27a] :
( ( V0E @ V1x @ V2y )
<=> ( ( V0E @ V1x )
= ( V0E @ V2y ) ) ) ) ).
thf(thm_2Equotient_2EPARTIAL__EQUIV__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Equotient_2EPARTIAL__EQUIV @ A_27a @ V0R )
<=> ( ? [V1x: A_27a] : ( V0R @ V1x @ V1x )
& ! [V2x: A_27a,V3y: A_27a] :
( ( V0R @ V2x @ V3y )
<=> ( ( V0R @ V2x @ V2x )
& ( V0R @ V3y @ V3y )
& ( ( V0R @ V2x )
= ( V0R @ V3y ) ) ) ) ) ) ).
thf(thm_2Equotient_2EQUOTIENT__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
<=> ( ! [V3a: A_27b] :
( ( V1abs @ ( V2rep @ V3a ) )
= V3a )
& ! [V4a: A_27b] : ( V0R @ ( V2rep @ V4a ) @ ( V2rep @ V4a ) )
& ! [V5r: A_27a,V6s: A_27a] :
( ( V0R @ V5r @ V6s )
<=> ( ( V0R @ V5r @ V5r )
& ( V0R @ V6s @ V6s )
& ( ( V1abs @ V5r )
= ( V1abs @ V6s ) ) ) ) ) ) ).
thf(thm_2Equotient_2EFUN__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f: A_27a > A_27c,V1g: A_27b > A_27d] :
( ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27b @ A_27c @ A_27d @ V0f @ V1g )
= ( ^ [V2h: A_27c > A_27b,V3x: A_27a] : ( V1g @ ( V2h @ ( V0f @ V3x ) ) ) ) ) ).
thf(thm_2Equotient_2EFUN__REL,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o,V2f: A_27a > A_27b,V3g: A_27a > A_27b] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V1R2 @ V2f @ V3g )
<=> ! [V4x: A_27a,V5y: A_27a] :
( ( V0R1 @ V4x @ V5y )
=> ( V1R2 @ ( V2f @ V4x ) @ ( V3g @ V5y ) ) ) ) ).
thf(thm_2Equotient_2Erespects__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Equotient_2Erespects @ A_27a @ A_27b )
= ( c_2Ecombin_2EW @ A_27a @ A_27b ) ) ).
thf(thm_2Equotient_2E_3F_21_21,axiom,
! [A_27a: $tType,V0P: A_27a > $o] :
( ( c_2Equotient_2E_3F_21_21 @ A_27a @ V0P )
= ( c_2Ebool_2E_3F_21 @ A_27a @ V0P ) ) ).
thf(thm_2Equotient_2ERES__EXISTS__EQUIV__DEF,axiom,
! [A_27a: $tType] :
( ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a )
= ( ^ [V0R: A_27a > A_27a > $o,V1P: A_27a > $o] :
( c_2Ebool_2E_2F_5C
@ ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R )
@ ^ [V2x: A_27a] : ( V1P @ V2x ) )
@ ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R )
@ ^ [V3x: A_27a] :
( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R )
@ ^ [V4y: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( V1P @ V3x ) @ ( V1P @ V4y ) ) @ ( V0R @ V3x @ V4y ) ) ) ) ) ) ) ).
thf(thm_2Equotient_2EFUN__REL__EQUALS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27a > A_27b,V7g: A_27a > A_27b] :
( ( ( c_2Equotient_2Erespects @ ( A_27a > A_27b ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 ) @ V6f )
& ( c_2Equotient_2Erespects @ ( A_27a > A_27b ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 ) @ V7g ) )
=> ( ( ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d @ V2rep1 @ V4abs2 @ V6f )
= ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d @ V2rep1 @ V4abs2 @ V7g ) )
<=> ! [V8x: A_27a,V9y: A_27a] :
( ( V0R1 @ V8x @ V9y )
=> ( V3R2 @ ( V6f @ V8x ) @ ( V7g @ V9y ) ) ) ) ) ) ) ).
thf(thm_2Equotient_2EEQUIV__IMP__PARTIAL__EQUIV,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Equotient_2EEQUIV @ A_27a @ V0R )
=> ( c_2Equotient_2EPARTIAL__EQUIV @ A_27a @ V0R ) ) ).
thf(thm_2Equotient_2EQUOTIENT__ABS__REP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3a: A_27b] :
( ( V1abs @ ( V2rep @ V3a ) )
= V3a ) ) ).
thf(thm_2Equotient_2EQUOTIENT__REP__REFL,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3a: A_27b] : ( V0R @ ( V2rep @ V3a ) @ ( V2rep @ V3a ) ) ) ).
thf(thm_2Equotient_2EQUOTIENT__REL,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3r: A_27a,V4s: A_27a] :
( ( V0R @ V3r @ V4s )
<=> ( ( V0R @ V3r @ V3r )
& ( V0R @ V4s @ V4s )
& ( ( V1abs @ V3r )
= ( V1abs @ V4s ) ) ) ) ) ).
thf(thm_2Equotient_2EQUOTIENT__REL__ABS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3r: A_27a,V4s: A_27a] :
( ( V0R @ V3r @ V4s )
=> ( ( V1abs @ V3r )
= ( V1abs @ V4s ) ) ) ) ).
thf(thm_2Equotient_2EQUOTIENT__REL__ABS__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3r: A_27a,V4s: A_27a] :
( ( V0R @ V3r @ V3r )
=> ( ( V0R @ V4s @ V4s )
=> ( ( V0R @ V3r @ V4s )
<=> ( ( V1abs @ V3r )
= ( V1abs @ V4s ) ) ) ) ) ) ).
thf(thm_2Equotient_2EQUOTIENT__REL__REP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3a: A_27b,V4b: A_27b] :
( ( V0R @ ( V2rep @ V3a ) @ ( V2rep @ V4b ) )
<=> ( V3a = V4b ) ) ) ).
thf(thm_2Equotient_2EQUOTIENT__REP__ABS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3r: A_27a] :
( ( V0R @ V3r @ V3r )
=> ( V0R @ ( V2rep @ ( V1abs @ V3r ) ) @ V3r ) ) ) ).
thf(thm_2Equotient_2EIDENTITY__EQUIV,axiom,
! [A_27a: $tType] : ( c_2Equotient_2EEQUIV @ A_27a @ ( c_2Emin_2E_3D @ A_27a ) ) ).
thf(thm_2Equotient_2EIDENTITY__QUOTIENT,axiom,
! [A_27a: $tType] : ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27a @ ( c_2Emin_2E_3D @ A_27a ) @ ( c_2Ecombin_2EI @ A_27a ) @ ( c_2Ecombin_2EI @ A_27a ) ) ).
thf(thm_2Equotient_2EEQUIV__REFL__SYM__TRANS,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ! [V1x: A_27a,V2y: A_27a] :
( ( V0R @ V1x @ V2y )
<=> ( ( V0R @ V1x )
= ( V0R @ V2y ) ) )
<=> ( ! [V3x: A_27a] : ( V0R @ V3x @ V3x )
& ! [V4x: A_27a,V5y: A_27a] :
( ( V0R @ V4x @ V5y )
=> ( V0R @ V5y @ V4x ) )
& ! [V6x: A_27a,V7y: A_27a,V8z: A_27a] :
( ( ( V0R @ V6x @ V7y )
& ( V0R @ V7y @ V8z ) )
=> ( V0R @ V6x @ V8z ) ) ) ) ).
thf(thm_2Equotient_2EQUOTIENT__SYM,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3x: A_27a,V4y: A_27a] :
( ( V0R @ V3x @ V4y )
=> ( V0R @ V4y @ V3x ) ) ) ).
thf(thm_2Equotient_2EQUOTIENT__TRANS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3x: A_27a,V4y: A_27a,V5z: A_27a] :
( ( ( V0R @ V3x @ V4y )
& ( V0R @ V4y @ V5z ) )
=> ( V0R @ V3x @ V5z ) ) ) ).
thf(thm_2Equotient_2EFUN__MAP__THM,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0f: A_27a > A_27c,V1g: A_27b > A_27d,V2h: A_27c > A_27b,V3x: A_27a] :
( ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27b @ A_27c @ A_27d @ V0f @ V1g @ V2h @ V3x )
= ( V1g @ ( V2h @ ( V0f @ V3x ) ) ) ) ).
thf(thm_2Equotient_2EFUN__MAP__I,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27b @ A_27a @ A_27b @ ( c_2Ecombin_2EI @ A_27a ) @ ( c_2Ecombin_2EI @ A_27b ) )
= ( c_2Ecombin_2EI @ ( A_27a > A_27b ) ) ) ).
thf(thm_2Equotient_2EIN__FUN,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1g: $o > $o,V2s: A_27b > $o,V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V0f @ V1g @ V2s ) )
= ( V1g @ ( c_2Ebool_2EIN @ A_27b @ ( V0f @ V3x ) @ V2s ) ) ) ).
thf(thm_2Equotient_2EFUN__REL__EQ,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ ( c_2Emin_2E_3D @ A_27a ) @ ( c_2Emin_2E_3D @ A_27b ) )
= ( c_2Emin_2E_3D @ ( A_27a > A_27b ) ) ) ).
thf(thm_2Equotient_2EFUN__QUOTIENT,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ( c_2Equotient_2EQUOTIENT @ ( A_27a > A_27b ) @ ( A_27c > A_27d ) @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d @ V2rep1 @ V4abs2 ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b @ V1abs1 @ V5rep2 ) ) ) ) ).
thf(thm_2Equotient_2ERESPECTS,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a] :
( ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R @ V1x )
= ( V0R @ V1x @ V1x ) ) ).
thf(thm_2Equotient_2EIN__RESPECTS,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R ) )
= ( V0R @ V1x @ V1x ) ) ).
thf(thm_2Equotient_2ERESPECTS__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o,V2f: A_27a > A_27b] :
( ( c_2Equotient_2Erespects @ ( A_27a > A_27b ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V1R2 ) @ V2f )
<=> ! [V3x: A_27a,V4y: A_27a] :
( ( V0R1 @ V3x @ V4y )
=> ( V1R2 @ ( V2f @ V3x ) @ ( V2f @ V4y ) ) ) ) ).
thf(thm_2Equotient_2ERESPECTS__MP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o,V2f: A_27a > A_27b,V3x: A_27a,V4y: A_27a] :
( ( ( c_2Equotient_2Erespects @ ( A_27a > A_27b ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V1R2 ) @ V2f )
& ( V0R1 @ V3x @ V4y ) )
=> ( V1R2 @ ( V2f @ V3x ) @ ( V2f @ V4y ) ) ) ).
thf(thm_2Equotient_2ERESPECTS__REP__ABS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4f: A_27a > A_27b,V5x: A_27a] :
( ( ( c_2Equotient_2Erespects @ ( A_27a > A_27b ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 ) @ V4f )
& ( V0R1 @ V5x @ V5x ) )
=> ( V3R2 @ ( V4f @ ( V2rep1 @ ( V1abs1 @ V5x ) ) ) @ ( V4f @ V5x ) ) ) ) ).
thf(thm_2Equotient_2ERESPECTS__o,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o,V2R3: A_27c > A_27c > $o,V3f: A_27b > A_27c,V4g: A_27a > A_27b] :
( ( ( c_2Equotient_2Erespects @ ( A_27b > A_27c ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ A_27c @ V1R2 @ V2R3 ) @ V3f )
& ( c_2Equotient_2Erespects @ ( A_27a > A_27b ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V1R2 ) @ V4g ) )
=> ( c_2Equotient_2Erespects @ ( A_27a > A_27c ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27c @ V0R1 @ V2R3 ) @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V3f @ V4g ) ) ) ).
thf(thm_2Equotient_2ERES__EXISTS__EQUIV,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1m: A_27a > $o] :
( ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a @ V0R @ V1m )
<=> ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R )
@ ^ [V2x: A_27a] : ( V1m @ V2x ) )
& ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R )
@ ^ [V3x: A_27a] :
( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R )
@ ^ [V4y: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( V1m @ V3x ) @ ( V1m @ V4y ) ) @ ( V0R @ V3x @ V4y ) ) ) ) ) ) ).
thf(thm_2Equotient_2EFUN__REL__EQ__REL,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27a > A_27b,V7g: A_27a > A_27b] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V6f @ V7g )
<=> ( ( c_2Equotient_2Erespects @ ( A_27a > A_27b ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 ) @ V6f )
& ( c_2Equotient_2Erespects @ ( A_27a > A_27b ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 ) @ V7g )
& ( ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d @ V2rep1 @ V4abs2 @ V6f )
= ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d @ V2rep1 @ V4abs2 @ V7g ) ) ) ) ) ) ).
thf(thm_2Equotient_2EFUN__REL__MP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27a > A_27b,V7g: A_27a > A_27b,V8x: A_27a,V9y: A_27a] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V6f @ V7g )
& ( V0R1 @ V8x @ V9y ) )
=> ( V3R2 @ ( V6f @ V8x ) @ ( V7g @ V9y ) ) ) ) ) ).
thf(thm_2Equotient_2EFUN__REL__IMP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27a > A_27b,V7g: A_27a > A_27b] :
( ( ( c_2Equotient_2Erespects @ ( A_27a > A_27b ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 ) @ V6f )
& ( c_2Equotient_2Erespects @ ( A_27a > A_27b ) @ $o @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 ) @ V7g )
& ( ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d @ V2rep1 @ V4abs2 @ V6f )
= ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d @ V2rep1 @ V4abs2 @ V7g ) ) )
=> ! [V8x: A_27a,V9y: A_27a] :
( ( V0R1 @ V8x @ V9y )
=> ( V3R2 @ ( V6f @ V8x ) @ ( V7g @ V9y ) ) ) ) ) ) ).
thf(thm_2Equotient_2EEQUALS__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3x: A_27b,V4y: A_27b] :
( ( V3x = V4y )
<=> ( V0R @ ( V2rep @ V3x ) @ ( V2rep @ V4y ) ) ) ) ).
thf(thm_2Equotient_2EEQUALS__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3x1: A_27a,V4x2: A_27a,V5y1: A_27a,V6y2: A_27a] :
( ( ( V0R @ V3x1 @ V4x2 )
& ( V0R @ V5y1 @ V6y2 ) )
=> ( ( V0R @ V3x1 @ V5y1 )
= ( V0R @ V4x2 @ V6y2 ) ) ) ) ).
thf(thm_2Equotient_2ELAMBDA__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27c > A_27d] :
( ( ^ [V7x: A_27c] : ( V6f @ V7x ) )
= ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d @ V2rep1 @ V4abs2
@ ^ [V8x: A_27a] : ( V5rep2 @ ( V6f @ ( V1abs1 @ V8x ) ) ) ) ) ) ) ).
thf(thm_2Equotient_2ELAMBDA__PRS1,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27c > A_27d] :
( ( ^ [V7x: A_27c] : ( V6f @ V7x ) )
= ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d @ V2rep1 @ V4abs2
@ ^ [V8x: A_27a] : ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b @ V1abs1 @ V5rep2 @ V6f @ V8x ) ) ) ) ) ).
thf(thm_2Equotient_2ELAMBDA__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f1: A_27a > A_27b,V7f2: A_27a > A_27b] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V6f1 @ V7f2 )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2
@ ^ [V8x: A_27a] : ( V6f1 @ V8x )
@ ^ [V9y: A_27a] : ( V7f2 @ V9y ) ) ) ) ) ).
thf(thm_2Equotient_2EABSTRACT__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27c > A_27d] :
( V6f
= ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d @ V2rep1 @ V4abs2 @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R1 ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b @ V1abs1 @ V5rep2 @ V6f ) ) ) ) ) ) ).
thf(thm_2Equotient_2ERES__ABSTRACT__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f1: A_27a > A_27b,V7f2: A_27a > A_27b] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V6f1 @ V7f2 )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R1 ) @ V6f1 ) @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R1 ) @ V7f2 ) ) ) ) ) ).
thf(thm_2Equotient_2ELET__RES__ABSTRACT,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: A_27a > $o,V1lam: A_27a > A_27b,V2v: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2v @ V0r )
=> ( ( c_2Ebool_2ELET @ A_27a @ A_27b @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ V0r @ V1lam ) @ V2v )
= ( c_2Ebool_2ELET @ A_27a @ A_27b @ V1lam @ V2v ) ) ) ).
thf(thm_2Equotient_2ELAMBDA__REP__ABS__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0REL1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a,V3REL2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b,V6f1: A_27a > A_27b,V7f2: A_27a > A_27b] :
( ( ! [V8r: A_27a,V9r_27: A_27a] :
( ( V0REL1 @ V8r @ V9r_27 )
=> ( V0REL1 @ V8r @ ( V2rep1 @ ( V1abs1 @ V9r_27 ) ) ) )
& ! [V10r: A_27b,V11r_27: A_27b] :
( ( V3REL2 @ V10r @ V11r_27 )
=> ( V3REL2 @ V10r @ ( V5rep2 @ ( V4abs2 @ V11r_27 ) ) ) )
& ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0REL1 @ V3REL2 @ V6f1 @ V7f2 ) )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0REL1 @ V3REL2 @ V6f1 @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b @ V1abs1 @ V5rep2 @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d @ V2rep1 @ V4abs2 @ V7f2 ) ) ) ) ).
thf(thm_2Equotient_2EREP__ABS__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0REL: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0REL @ V1abs @ V2rep )
=> ! [V3x1: A_27a,V4x2: A_27a] :
( ( V0REL @ V3x1 @ V4x2 )
=> ( V0REL @ V3x1 @ ( V2rep @ ( V1abs @ V4x2 ) ) ) ) ) ).
thf(thm_2Equotient_2EFORALL__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3f: A_27b > $o] :
( ( c_2Ebool_2E_21 @ A_27b @ V3f )
= ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3f ) ) ) ) ).
thf(thm_2Equotient_2ERES__FORALL__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3f: A_27a > $o,V4g: A_27a > $o] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V3f @ V4g )
=> ( ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R ) @ V3f )
= ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R ) @ V4g ) ) ) ) ).
thf(thm_2Equotient_2ERES__FORALL__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3P: A_27b > $o,V4f: A_27b > $o] :
( ( c_2Ebool_2ERES__FORALL @ A_27b @ V3P @ V4f )
= ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3P ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V4f ) ) ) ) ).
thf(thm_2Equotient_2EEXISTS__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3f: A_27b > $o] :
( ( c_2Ebool_2E_3F @ A_27b @ V3f )
= ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3f ) ) ) ) ).
thf(thm_2Equotient_2ERES__EXISTS__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3f: A_27a > $o,V4g: A_27a > $o] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V3f @ V4g )
=> ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R ) @ V3f )
= ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R ) @ V4g ) ) ) ) ).
thf(thm_2Equotient_2ERES__EXISTS__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3P: A_27b > $o,V4f: A_27b > $o] :
( ( c_2Ebool_2ERES__EXISTS @ A_27b @ V3P @ V4f )
= ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3P ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V4f ) ) ) ) ).
thf(thm_2Equotient_2EEXISTS__UNIQUE__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3f: A_27b > $o] :
( ( c_2Ebool_2E_3F_21 @ A_27b @ V3f )
= ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a @ V0R @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ $o @ A_27b @ $o @ V1abs @ ( c_2Ecombin_2EI @ $o ) @ V3f ) ) ) ) ).
thf(thm_2Equotient_2ERES__EXISTS__EQUIV__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3f: A_27a > $o,V4g: A_27a > $o] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V3f @ V4g )
=> ( ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a @ V0R @ V3f )
= ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a @ V0R @ V4g ) ) ) ) ).
thf(thm_2Equotient_2ECOND__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3a: $o,V4b: A_27b,V5c: A_27b] :
( ( c_2Ebool_2ECOND @ A_27b @ V3a @ V4b @ V5c )
= ( V1abs @ ( c_2Ebool_2ECOND @ A_27a @ V3a @ ( V2rep @ V4b ) @ ( V2rep @ V5c ) ) ) ) ) ).
thf(thm_2Equotient_2ECOND__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3a1: $o,V4a2: $o,V5b1: A_27a,V6b2: A_27a,V7c1: A_27a,V8c2: A_27a] :
( ( ( V3a1 = V4a2 )
& ( V0R @ V5b1 @ V6b2 )
& ( V0R @ V7c1 @ V8c2 ) )
=> ( V0R @ ( c_2Ebool_2ECOND @ A_27a @ V3a1 @ V5b1 @ V7c1 ) @ ( c_2Ebool_2ECOND @ A_27a @ V4a2 @ V6b2 @ V8c2 ) ) ) ) ).
thf(thm_2Equotient_2ELET__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27c > A_27d,V7x: A_27c] :
( ( c_2Ebool_2ELET @ A_27c @ A_27d @ V6f @ V7x )
= ( V4abs2 @ ( c_2Ebool_2ELET @ A_27a @ A_27b @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b @ V1abs1 @ V5rep2 @ V6f ) @ ( V2rep1 @ V7x ) ) ) ) ) ) ).
thf(thm_2Equotient_2ELET__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27a > A_27b,V7g: A_27a > A_27b,V8x: A_27a,V9y: A_27a] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V6f @ V7g )
& ( V0R1 @ V8x @ V9y ) )
=> ( V3R2 @ ( c_2Ebool_2ELET @ A_27a @ A_27b @ V6f @ V8x ) @ ( c_2Ebool_2ELET @ A_27a @ A_27b @ V7g @ V9y ) ) ) ) ) ).
thf(thm_2Equotient_2Eliteral__case__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27c > A_27d,V7x: A_27c] :
( ( c_2Ebool_2Eliteral__case @ A_27c @ A_27d @ V6f @ V7x )
= ( V4abs2 @ ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b @ V1abs1 @ V5rep2 @ V6f ) @ ( V2rep1 @ V7x ) ) ) ) ) ) ).
thf(thm_2Equotient_2Eliteral__case__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27a > A_27b,V7g: A_27a > A_27b,V8x: A_27a,V9y: A_27a] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V6f @ V7g )
& ( V0R1 @ V8x @ V9y ) )
=> ( V3R2 @ ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b @ V6f @ V8x ) @ ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b @ V7g @ V9y ) ) ) ) ) ).
thf(thm_2Equotient_2EAPPLY__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27c > A_27d,V7x: A_27c] :
( ( V6f @ V7x )
= ( V4abs2 @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b @ V1abs1 @ V5rep2 @ V6f @ ( V2rep1 @ V7x ) ) ) ) ) ) ).
thf(thm_2Equotient_2EAPPLY__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27a > A_27b,V7g: A_27a > A_27b,V8x: A_27a,V9y: A_27a] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V6f @ V7g )
& ( V0R1 @ V8x @ V9y ) )
=> ( V3R2 @ ( V6f @ V8x ) @ ( V7g @ V9y ) ) ) ) ) ).
thf(thm_2Equotient_2EI__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3e: A_27b] :
( ( c_2Ecombin_2EI @ A_27b @ V3e )
= ( V1abs @ ( c_2Ecombin_2EI @ A_27a @ ( V2rep @ V3e ) ) ) ) ) ).
thf(thm_2Equotient_2EI__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27a > $o,V1abs: A_27a > A_27b,V2rep: A_27b > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b @ V0R @ V1abs @ V2rep )
=> ! [V3e1: A_27a,V4e2: A_27a] :
( ( V0R @ V3e1 @ V4e2 )
=> ( V0R @ ( c_2Ecombin_2EI @ A_27a @ V3e1 ) @ ( c_2Ecombin_2EI @ A_27a @ V4e2 ) ) ) ) ).
thf(thm_2Equotient_2EK__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6x: A_27c,V7y: A_27d] :
( ( c_2Ecombin_2EK @ A_27c @ A_27d @ V6x @ V7y )
= ( V1abs1 @ ( c_2Ecombin_2EK @ A_27a @ A_27b @ ( V2rep1 @ V6x ) @ ( V5rep2 @ V7y ) ) ) ) ) ) ).
thf(thm_2Equotient_2EK__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6x1: A_27a,V7x2: A_27a,V8y1: A_27b,V9y2: A_27b] :
( ( ( V0R1 @ V6x1 @ V7x2 )
& ( V3R2 @ V8y1 @ V9y2 ) )
=> ( V0R1 @ ( c_2Ecombin_2EK @ A_27a @ A_27b @ V6x1 @ V8y1 ) @ ( c_2Ecombin_2EK @ A_27a @ A_27b @ V7x2 @ V9y2 ) ) ) ) ) ).
thf(thm_2Equotient_2Eo__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27d,V2rep1: A_27d > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27e,V5rep2: A_27e > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6R3: A_27c > A_27c > $o,V7abs3: A_27c > A_27f,V8rep3: A_27f > A_27c] :
( ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f @ V6R3 @ V7abs3 @ V8rep3 )
=> ! [V9f: A_27e > A_27f,V10g: A_27d > A_27e] :
( ( c_2Ecombin_2Eo @ A_27d @ A_27f @ A_27e @ V9f @ V10g )
= ( c_2Equotient_2E_2D_2D_3E @ A_27d @ A_27c @ A_27a @ A_27f @ V2rep1 @ V7abs3 @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ A_27f @ A_27e @ A_27c @ V4abs2 @ V8rep3 @ V9f ) @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27e @ A_27d @ A_27b @ V1abs1 @ V5rep2 @ V10g ) ) ) ) ) ) ) ).
thf(thm_2Equotient_2Eo__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27d,V2rep1: A_27d > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27e,V5rep2: A_27e > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6R3: A_27c > A_27c > $o,V7abs3: A_27c > A_27f,V8rep3: A_27f > A_27c] :
( ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f @ V6R3 @ V7abs3 @ V8rep3 )
=> ! [V9f1: A_27b > A_27c,V10f2: A_27b > A_27c,V11g1: A_27a > A_27b,V12g2: A_27a > A_27b] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ A_27c @ V3R2 @ V6R3 @ V9f1 @ V10f2 )
& ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V11g1 @ V12g2 ) )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27c @ V0R1 @ V6R3 @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V9f1 @ V11g1 ) @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b @ V10f2 @ V12g2 ) ) ) ) ) ) ).
thf(thm_2Equotient_2EC__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27d,V2rep1: A_27d > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27e,V5rep2: A_27e > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6R3: A_27c > A_27c > $o,V7abs3: A_27c > A_27f,V8rep3: A_27f > A_27c] :
( ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f @ V6R3 @ V7abs3 @ V8rep3 )
=> ! [V9f: A_27d > A_27e > A_27f,V10x: A_27e,V11y: A_27d] :
( ( c_2Ecombin_2EC @ A_27d @ A_27e @ A_27f @ V9f @ V10x @ V11y )
= ( V7abs3 @ ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27c @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ ( A_27e > A_27f ) @ A_27d @ ( A_27b > A_27c ) @ V1abs1 @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ A_27f @ A_27e @ A_27c @ V4abs2 @ V8rep3 ) @ V9f ) @ ( V5rep2 @ V10x ) @ ( V2rep1 @ V11y ) ) ) ) ) ) ) ).
thf(thm_2Equotient_2EC__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,A_27f: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27d,V2rep1: A_27d > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27e,V5rep2: A_27e > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6R3: A_27c > A_27c > $o,V7abs3: A_27c > A_27f,V8rep3: A_27f > A_27c] :
( ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f @ V6R3 @ V7abs3 @ V8rep3 )
=> ! [V9f1: A_27a > A_27b > A_27c,V10f2: A_27a > A_27b > A_27c,V11x1: A_27b,V12x2: A_27b,V13y1: A_27a,V14y2: A_27a] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ ( A_27b > A_27c ) @ V0R1 @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ A_27c @ V3R2 @ V6R3 ) @ V9f1 @ V10f2 )
& ( V3R2 @ V11x1 @ V12x2 )
& ( V0R1 @ V13y1 @ V14y2 ) )
=> ( V6R3 @ ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27c @ V9f1 @ V11x1 @ V13y1 ) @ ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27c @ V10f2 @ V12x2 @ V14y2 ) ) ) ) ) ) ).
thf(thm_2Equotient_2EW__PRS,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f: A_27c > A_27c > A_27d,V7x: A_27c] :
( ( c_2Ecombin_2EW @ A_27c @ A_27d @ V6f @ V7x )
= ( V4abs2 @ ( c_2Ecombin_2EW @ A_27a @ A_27b @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ ( A_27c > A_27d ) @ A_27c @ ( A_27a > A_27b ) @ V1abs1 @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b @ V1abs1 @ V5rep2 ) @ V6f ) @ ( V2rep1 @ V7x ) ) ) ) ) ) ).
thf(thm_2Equotient_2EW__RSP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4abs2: A_27b > A_27d,V5rep2: A_27d > A_27b] :
( ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d @ V3R2 @ V4abs2 @ V5rep2 )
=> ! [V6f1: A_27a > A_27a > A_27b,V7f2: A_27a > A_27a > A_27b,V8x1: A_27a,V9x2: A_27a] :
( ( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ ( A_27a > A_27b ) @ V0R1 @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 ) @ V6f1 @ V7f2 )
& ( V0R1 @ V8x1 @ V9x2 ) )
=> ( V3R2 @ ( c_2Ecombin_2EW @ A_27a @ A_27b @ V6f1 @ V8x1 ) @ ( c_2Ecombin_2EW @ A_27a @ A_27b @ V7f2 @ V9x2 ) ) ) ) ) ).
thf(thm_2Equotient_2EEQ__IMPLIES,axiom,
! [V0P: $o,V1Q: $o] :
( ( V0P = V1Q )
=> ( V0P
=> V1Q ) ) ).
thf(thm_2Equotient_2EEQUALS__IMPLIES,axiom,
! [A_27a: $tType,V0P: A_27a,V1P_27: A_27a,V2Q: A_27a,V3Q_27: A_27a] :
( ( ( V0P = V2Q )
& ( V1P_27 = V3Q_27 ) )
=> ( ( V0P = V1P_27 )
=> ( V2Q = V3Q_27 ) ) ) ).
thf(thm_2Equotient_2ECONJ__IMPLIES,axiom,
! [V0P: $o,V1P_27: $o,V2Q: $o,V3Q_27: $o] :
( ( ( V0P
=> V2Q )
& ( V1P_27
=> V3Q_27 ) )
=> ( ( V0P
& V1P_27 )
=> ( V2Q
& V3Q_27 ) ) ) ).
thf(thm_2Equotient_2EDISJ__IMPLIES,axiom,
! [V0P: $o,V1P_27: $o,V2Q: $o,V3Q_27: $o] :
( ( ( V0P
=> V2Q )
& ( V1P_27
=> V3Q_27 ) )
=> ( ( V0P
| V1P_27 )
=> ( V2Q
| V3Q_27 ) ) ) ).
thf(thm_2Equotient_2EIMP__IMPLIES,axiom,
! [V0P: $o,V1P_27: $o,V2Q: $o,V3Q_27: $o] :
( ( ( V2Q
=> V0P )
& ( V1P_27
=> V3Q_27 ) )
=> ( ( V0P
=> V1P_27 )
=> ( V2Q
=> V3Q_27 ) ) ) ).
thf(thm_2Equotient_2ENOT__IMPLIES,axiom,
! [V0P: $o,V1Q: $o] :
( ( V1Q
=> V0P )
=> ( ( (~) @ V0P )
=> ( (~) @ V1Q ) ) ) ).
thf(thm_2Equotient_2EEQUALS__EQUIV__IMPLIES,axiom,
! [A_27a: $tType,V0b2: A_27a,V1b1: A_27a,V2a2: A_27a,V3a1: A_27a,V4R: A_27a > A_27a > $o] :
( ( c_2Equotient_2EEQUIV @ A_27a @ V4R )
=> ( ( ( V4R @ V3a1 @ V2a2 )
& ( V4R @ V1b1 @ V0b2 ) )
=> ( ( V3a1 = V1b1 )
=> ( V4R @ V2a2 @ V0b2 ) ) ) ) ).
thf(thm_2Equotient_2EABSTRACT__RES__ABSTRACT,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4f: A_27a > A_27b,V5g: A_27a > A_27b] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V4f @ V5g )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V4f @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R1 ) @ V5g ) ) ) ) ).
thf(thm_2Equotient_2ERES__ABSTRACT__ABSTRACT,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0R1: A_27a > A_27a > $o,V1abs1: A_27a > A_27c,V2rep1: A_27c > A_27a] :
( ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c @ V0R1 @ V1abs1 @ V2rep1 )
=> ! [V3R2: A_27b > A_27b > $o,V4f: A_27a > A_27b,V5g: A_27a > A_27b] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ V4f @ V5g )
=> ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b @ V0R1 @ V3R2 @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R1 ) @ V4f ) @ V5g ) ) ) ).
thf(thm_2Equotient_2EEQUIV__RES__ABSTRACT__LEFT,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o,V2f1: A_27a > A_27b,V3f2: A_27a > A_27b,V4x1: A_27a,V5x2: A_27a] :
( ( ( V1R2 @ ( V2f1 @ V4x1 ) @ ( V3f2 @ V5x2 ) )
& ( V0R1 @ V4x1 @ V4x1 ) )
=> ( V1R2 @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R1 ) @ V2f1 @ V4x1 ) @ ( V3f2 @ V5x2 ) ) ) ).
thf(thm_2Equotient_2EEQUIV__RES__ABSTRACT__RIGHT,axiom,
! [A_27a: $tType,A_27b: $tType,V0R1: A_27a > A_27a > $o,V1R2: A_27b > A_27b > $o,V2f1: A_27a > A_27b,V3f2: A_27a > A_27b,V4x1: A_27a,V5x2: A_27a] :
( ( ( V1R2 @ ( V2f1 @ V4x1 ) @ ( V3f2 @ V5x2 ) )
& ( V0R1 @ V5x2 @ V5x2 ) )
=> ( V1R2 @ ( V2f1 @ V4x1 ) @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R1 ) @ V3f2 @ V5x2 ) ) ) ).
thf(thm_2Equotient_2EEQUIV__RES__FORALL,axiom,
! [A_27a: $tType,V0E: A_27a > A_27a > $o,V1P: A_27a > $o] :
( ( c_2Equotient_2EEQUIV @ A_27a @ V0E )
=> ( ( c_2Ebool_2ERES__FORALL @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0E ) @ V1P )
= ( c_2Ebool_2E_21 @ A_27a @ V1P ) ) ) ).
thf(thm_2Equotient_2EEQUIV__RES__EXISTS,axiom,
! [A_27a: $tType,V0E: A_27a > A_27a > $o,V1P: A_27a > $o] :
( ( c_2Equotient_2EEQUIV @ A_27a @ V0E )
=> ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0E ) @ V1P )
= ( c_2Ebool_2E_3F @ A_27a @ V1P ) ) ) ).
thf(thm_2Equotient_2EEQUIV__RES__EXISTS__UNIQUE,axiom,
! [A_27a: $tType,V0E: A_27a > A_27a > $o,V1P: A_27a > $o] :
( ( c_2Equotient_2EEQUIV @ A_27a @ V0E )
=> ( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0E ) @ V1P )
= ( c_2Ebool_2E_3F_21 @ A_27a @ V1P ) ) ) ).
thf(thm_2Equotient_2EFORALL__REGULAR,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
( ! [V2x: A_27a] :
( ( V0P @ V2x )
=> ( V1Q @ V2x ) )
=> ( ( c_2Ebool_2E_21 @ A_27a @ V0P )
=> ( c_2Ebool_2E_21 @ A_27a @ V1Q ) ) ) ).
thf(thm_2Equotient_2EEXISTS__REGULAR,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o] :
( ! [V2x: A_27a] :
( ( V0P @ V2x )
=> ( V1Q @ V2x ) )
=> ( ( c_2Ebool_2E_3F @ A_27a @ V0P )
=> ( c_2Ebool_2E_3F @ A_27a @ V1Q ) ) ) ).
thf(thm_2Equotient_2ERES__FORALL__REGULAR,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o,V2R: A_27a > $o] :
( ! [V3x: A_27a] :
( ( V2R @ V3x )
=> ( ( V0P @ V3x )
=> ( V1Q @ V3x ) ) )
=> ( ( c_2Ebool_2ERES__FORALL @ A_27a @ V2R @ V0P )
=> ( c_2Ebool_2ERES__FORALL @ A_27a @ V2R @ V1Q ) ) ) ).
thf(thm_2Equotient_2ERES__EXISTS__REGULAR,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1Q: A_27a > $o,V2R: A_27a > $o] :
( ! [V3x: A_27a] :
( ( V2R @ V3x )
=> ( ( V0P @ V3x )
=> ( V1Q @ V3x ) ) )
=> ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V2R @ V0P )
=> ( c_2Ebool_2ERES__EXISTS @ A_27a @ V2R @ V1Q ) ) ) ).
thf(thm_2Equotient_2ELEFT__RES__FORALL__REGULAR,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1R: A_27a > $o,V2Q: A_27a > $o] :
( ! [V3x: A_27a] :
( ( V1R @ V3x )
& ( ( V2Q @ V3x )
=> ( V0P @ V3x ) ) )
=> ( ( c_2Ebool_2ERES__FORALL @ A_27a @ V1R @ V2Q )
=> ( c_2Ebool_2E_21 @ A_27a @ V0P ) ) ) ).
thf(thm_2Equotient_2ERIGHT__RES__FORALL__REGULAR,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1R: A_27a > $o,V2Q: A_27a > $o] :
( ! [V3x: A_27a] :
( ( V1R @ V3x )
=> ( ( V0P @ V3x )
=> ( V2Q @ V3x ) ) )
=> ( ( c_2Ebool_2E_21 @ A_27a @ V0P )
=> ( c_2Ebool_2ERES__FORALL @ A_27a @ V1R @ V2Q ) ) ) ).
thf(thm_2Equotient_2ELEFT__RES__EXISTS__REGULAR,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1R: A_27a > $o,V2Q: A_27a > $o] :
( ! [V3x: A_27a] :
( ( V1R @ V3x )
=> ( ( V2Q @ V3x )
=> ( V0P @ V3x ) ) )
=> ( ( c_2Ebool_2ERES__EXISTS @ A_27a @ V1R @ V2Q )
=> ( c_2Ebool_2E_3F @ A_27a @ V0P ) ) ) ).
thf(thm_2Equotient_2ERIGHT__RES__EXISTS__REGULAR,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1R: A_27a > $o,V2Q: A_27a > $o] :
( ! [V3x: A_27a] :
( ( V1R @ V3x )
& ( ( V0P @ V3x )
=> ( V2Q @ V3x ) ) )
=> ( ( c_2Ebool_2E_3F @ A_27a @ V0P )
=> ( c_2Ebool_2ERES__EXISTS @ A_27a @ V1R @ V2Q ) ) ) ).
thf(thm_2Equotient_2EEXISTS__UNIQUE__REGULAR,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1E: A_27a > A_27a > $o,V2Q: A_27a > $o] :
( ( ! [V3x: A_27a] :
( ( V0P @ V3x )
=> ( ( c_2Equotient_2Erespects @ A_27a @ $o @ V1E @ V3x )
& ( V2Q @ V3x ) ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ( ( c_2Equotient_2Erespects @ A_27a @ $o @ V1E @ V4x )
& ( V2Q @ V4x )
& ( c_2Equotient_2Erespects @ A_27a @ $o @ V1E @ V5y )
& ( V2Q @ V5y ) )
=> ( V1E @ V4x @ V5y ) ) )
=> ( ( c_2Ebool_2E_3F_21 @ A_27a @ V0P )
=> ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a @ V1E @ V2Q ) ) ) ).
thf(thm_2Equotient_2ERES__EXISTS__UNIQUE__RESPECTS__REGULAR,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > $o] :
( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R ) @ V1P )
=> ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a @ V0R @ V1P ) ) ).
thf(thm_2Equotient_2ERES__EXISTS__UNIQUE__REGULAR,axiom,
! [A_27a: $tType,V0P: A_27a > $o,V1R: A_27a > A_27a > $o,V2Q: A_27a > $o] :
( ( ! [V3x: A_27a] :
( ( V0P @ V3x )
=> ( V2Q @ V3x ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ( ( c_2Equotient_2Erespects @ A_27a @ $o @ V1R @ V4x )
& ( V2Q @ V4x )
& ( c_2Equotient_2Erespects @ A_27a @ $o @ V1R @ V5y )
& ( V2Q @ V5y ) )
=> ( V1R @ V4x @ V5y ) ) )
=> ( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V1R ) @ V0P )
=> ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a @ V1R @ V2Q ) ) ) ).
thf(thm_2Equotient_2ERES__EXISTS__UNIQUE__REGULAR__SAME,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1P: A_27a > $o,V2Q: A_27a > $o] :
( ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ $o @ V0R @ ( c_2Emin_2E_3D @ $o ) @ V1P @ V2Q )
=> ( ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a @ ( c_2Equotient_2Erespects @ A_27a @ $o @ V0R ) @ V1P )
=> ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a @ V0R @ V2Q ) ) ) ).
%------------------------------------------------------------------------------